81.13.4 problem 17-4

Internal problem ID [21678]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 17. Reduction of Order. Page 420
Problem number : 17-4
Date solved : Thursday, October 02, 2025 at 07:59:50 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y-\frac {x y^{\prime }}{2}-\frac {x}{2 y^{\prime }}&=0 \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 26

ode:=y(x)-1/2*x*diff(y(x),x)-1/2*x/diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x \\ y &= x \\ y &= \frac {c_1^{2}+x^{2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.11 (sec). Leaf size: 41

ode=y[x]-x/2*D[y[x],x]-x/2*1/D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x \cosh (-\log (x)+c_1)\\ y(x)&\to -x \cosh (\log (x)+c_1)\\ y(x)&\to -x\\ y(x)&\to x \end{align*}

✓ Sympy. Time used: 3.901 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x)/2 - x/(2*Derivative(y(x), x)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = x \cosh {\left (C_{1} - \log {\left (x \right )} \right )}, \ y{\left (x \right )} = x \cosh {\left (C_{1} - \log {\left (x \right )} \right )}\right ] \]